Operator Theory

1. Operator
- 1.1. Adjoint
  - 1.1.1. Of Differential Operators
    - 1.1.1.1. Formal Adjoint
    - 1.1.1.2. Boundary Conditions
2. Fredholm-Alrternative Theorem
- 2.1. Statement
3. Operator Norm
4. Bounded Operator
5. Fundamental Solution
6. Operational Calculus
- 6.1. Shift
- 6.2. Shift Theorem
  - 6.2.1. Statement
7. Reference

1. Operator

General definition is not given.
It is often the mapping from a space of functions to another space of functions.

1.1. Adjoint

\[ \langle Ax, y\rangle = \langle x, A^*y\rangle \] for a given inner produce \( \langle\cdot,\cdot\rangle \).

1.1.1. Of Differential Operators

1.1.1.1. Formal Adjoint

It is the operator that is obtained by performing integration by part on the inner product.

1.1.1.2. Boundary Conditions

Boundary conditions on one of the operand must be given to constitute an adjoint, which restricts the space of the operand.

The boundary conditions of the other operand are chosen forming its own space, such that the conjunct or bilinear concomitant is zero: \[ J(u, v) := \langle v, Lu\rangle - \langle L^{\dagger} v, u\rangle = 0. \]

2. Fredholm-Alrternative Theorem

It is about the solvability of the \( Lu = f \).

2.1. Statement

Let \( A \) be an operator and \( x, y, b \) vectors, with \( Ax = b \) and \( A^*y= 0\).

Using the property of the adjoint, taking the inner product with \( y \) on the both side of \( Ax = b \) restricts \( b \) to be always orthogonal to the null space of \( A^* \), since the range of \( A \) is such.

Furthermore, if the null space of \( A^* \) is trivial, the equation is always solvable.

3. Operator Norm

It is the dual norm.
\[ \| T\| = \sup\left\{\frac{\|Tx\|_V}{\| x\|_U}\colon x \in U, x\neq 0\right\} \]

4. Bounded Operator

The operator norm is bounded.
An operator \(\mathrm{A}\colon U \to V\) for normed vector spaces \(U\) and \(V\), \(\mathrm{A}\) is bounded if: \[ \exists c > 0 \colon \| \mathrm{A}\mathbf{x}\|_V \le c\|\mathbf{x}\|_U. \]
Laplace transform is an integral operator.

5. Fundamental Solution

For a linear partial differential operator \( L \) , the fundamental solution \( F \) satisfies: \[ LF = \delta(x). \]
This is different formulation of the Green's function

6. Operational Calculus

Calculus with operators
The Abstract World of Operational Calculus - YouTube

6.1. Shift

\[ \exp\left(t\frac{\mathrm{d}}{\mathrm{d}x}\right) f(x) = f(x+t) \]

6.2. Shift Theorem

Exponential Shift Theorem

6.2.1. Statement

For a polynomial \(P\): \[ P(D)(e^{ax}y) = e^{ax}P(D+a)y. \]